% Advection diffusion with small viscosity
%
% Boundary conditions
%
%
T=1;
alpha=1;
beta=3;
%
m=100;
h=T/(m+1);
xx=[h:h:T-h]';
visc=0.005;
u=alpha+xx+(beta-alpha-1)*(exp(xx/visc)-1)./(exp(1/visc)-1);
plot([0;xx;T],[alpha;u;beta]);
e=ones(m,1);


A=1/(h^2)*spdiags([e -2*e e],-1:1,m,m);
%D=1/(2*h)*spdiags([-e e],[-1 1],m,m);
D=1/h*spdiags([-e e],-1:0,m,m);
F=-ones(m,1);
F(1,1)=F(1,1)-alpha*visc/(h^2)-alpha/(h);%alpha/(2*h);
F(end,1)=F(end,1)-beta*visc/(h^2);

Am=visc*A-D;
U=Am\F;
hold on
plot([0;xx;T],[alpha;U;beta],'r')
hold off